5.02a Discrete probability distributions: general

4. The discrete random variable \(X\) has the following probability distribution. Calculate

1. \(k\),
2. \(\mathrm { F } ( 2 )\),
3. \(\mathrm { P } ( 1.3 < X \leq 3.8 )\),
4. \(\mathrm { E } ( X )\),
5. \(\operatorname { Var } ( 3 X + 2 )\).

7. A bag contains 4 red and 2 blue balls, all of the same size. A ball is selected at random and removed from the bag. This is repeated until a blue ball is pulled out of the bag. The random variable \(B\) is the number of balls that have been removed from the bag.

1. Show that \(\mathrm { P } ( B = 2 ) = \frac { 4 } { 15 }\).
2. Find the probability distribution of \(B\).
3. Find \(\mathrm { E } ( B )\). The bag and the same 6 balls are used in a game at a funfair. One ball is removed from the bag at a time and a contestant wins 50 pence if one of the first two balls picked out is blue.
4. What are the expected winnings from playing this game once? For \(\pounds 1\), a contestant gets to play the game three times.
5. What is the expected profit or loss from the three games?

2. The discrete random variable \(X\) has the following probability distribution.

1. Find an expression for \(b\) in terms of \(a\).
2. Find an expression for \(\mathrm { E } ( X )\) in terms of \(a\). Given that \(\mathrm { E } ( X ) = \frac { 45 } { 16 }\),
3. find the values of \(a\) and \(b\),

5. A group of children were each asked to try and complete a task to test hand-eye coordination. Each child repeated the task until he or she had been successful or had made four attempts. The number of attempts made by the children in the group are summarised in the table below.

1. Calculate the mean and standard deviation of the number of attempts made by each child. It is suggested that the number of attempts made by each child could be modelled by a discrete random variable \(X\) with the probability function $$P ( X = x ) = \left\{ \begin{array} { c c } k \left( 20 - x ^ { 2 } \right) , & x = 1,2,3,4 \\ 0 , & \text { otherwise } \end{array} \right.$$
2. Show that \(k = \frac { 1 } { 50 }\).
3. Find \(\mathrm { E } ( X )\).
4. Comment on the suitability of this model.

1. The discrete random variable \(X\) has the following probability distribution.

1. Find and simplify an expression in terms of \(k\) for \(\mathrm { E } ( X )\). Given that \(\mathrm { E } ( X ) = 9\),
2. find the value of \(k\).
   

2. The discrete random variable \(X\) has the probability function shown below. $$P ( X = x ) = \left\{ \begin{array} { l c } \frac { k } { x } , & x = 1,2,3,4 \\ 0 , & \text { otherwise } . \end{array} \right.$$

1. Show that \(k = \frac { 12 } { 25 }\) Find
2. \(\mathrm { F } ( 2 )\),
3. \(\mathrm { E } ( X )\),
4. \(\mathrm { E } \left( X ^ { 2 } + 2 \right)\).

6. In a game two spinners are used. The score on the first spinner is given by the random variable \(A\), which has the following probability distribution:

1. State the name of this distribution.
2. Write down \(\mathrm { E } ( A )\). The score on the second spinner is given by the random variable \(B\), which has the following probability distribution:

   \(b\)	1	2	3
   \(\mathrm { P } ( B = b )\)	\(\frac { 1 } { 2 }\)	\(\frac { 1 } { 4 }\)	\(\frac { 1 } { 4 }\)

3. Find \(\mathrm { E } ( B )\). On each player's turn in the game, both spinners are used and the scores on the two spinners are added together. The total score on the two spinners is given by the random variable \(C\).
4. Show that \(\mathrm { P } ( C = 2 ) = \frac { 1 } { 6 }\).
5. Find the probability distribution of \(C\).
6. Show that \(\mathrm { E } ( C ) = \mathrm { E } ( A ) + \mathrm { E } ( B )\).

4 The number of fish, \(X\), caught by Pearl when she goes fishing can be modelled by the following discrete probability distribution:

1. Find the value of \(k\).
2. Find:
   1. \(\mathrm { E } ( X )\);
   2. \(\operatorname { Var } ( X )\).
3. When Pearl sells her fish, she earns a profit, in pounds, given by $$Y = 5 X + 2$$ Find:
   1. \(\mathrm { E } ( Y )\);
   2. the standard deviation of \(Y\).

5 A discrete random variable \(X\) has the probability distribution $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l } \frac { x } { 20 } & x = 1,2,3,4,5 \\ \frac { x } { 24 } & x = 6 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Calculate \(\mathrm { P } ( X \geqslant 5 )\).
2. 1. Show that \(\mathrm { E } \left( \frac { 1 } { X } \right) = \frac { 7 } { 24 }\).
   2. Hence, or otherwise, show that \(\operatorname { Var } \left( \frac { 1 } { X } \right) = 0.036\), correct to three decimal places.
3. Calculate the mean and the variance of \(A\), the area of rectangles having sides of length \(X + 3\) and \(\frac { 1 } { X }\).

6

1. Ali has a bag of 10 balls, of which 5 are red and 5 are blue. He asks Ben to select 5 of these balls from the bag at random. The probability distribution of \(X\), the number of red balls that Ben selects, is given in Table 1. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Table 1}

   \(\boldsymbol { x }\)	0	1	2	3	4	5
   \(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)	\(\frac { 1 } { 252 }\)	\(\frac { 25 } { 252 }\)	\(\frac { 100 } { 252 }\)	\(a\)	\(\frac { 25 } { 252 }\)	\(\frac { 1 } { 252 }\)

   \end{table}
   1. State the value of \(a\).
   2. Hence write down the value of \(\mathrm { E } ( X )\).
   3. Determine the standard deviation of \(X\).
2. Ali decides to play a game with Joanne using the same 10 balls. Joanne is asked to select 2 balls from the bag at random. Ali agrees to pay Joanne 90 p if the two balls that she selects are the same colour, but nothing if they are different colours. Joanne pays 50 p to play the game. The probability distribution of \(Y\), the number of red balls that Joanne selects, is given in Table 2. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Table 2}

   \(\boldsymbol { y }\)	0	1	2
   \(\mathbf { P } ( \boldsymbol { Y } = \boldsymbol { y } )\)	\(\frac { 2 } { 9 }\)	\(\frac { 5 } { 9 }\)	\(\frac { 2 } { 9 }\)

   \end{table}
   1. Determine whether Joanne can expect to make a profit or a loss from playing the game once.
   2. Hence calculate the expected size of this profit or loss after Joanne has played the game 100 times.
      (3 marks)

4

1. A red biased tetrahedral die is rolled. The number, \(X\), on the face on which it lands has the probability distribution given by

   \(\boldsymbol { x }\)	1	2	3	4
   \(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)	0.2	0.1	0.4	0.3

   1. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
   2. The red die is now rolled three times. The random variable \(S\) is the sum of the three numbers obtained. Find \(\mathrm { E } ( S )\) and \(\operatorname { Var } ( S )\).
2. A blue biased tetrahedral die is rolled. The number, \(Y\), on the face on which it lands has the probability distribution given by $$\mathrm { P } ( Y = y ) = \begin{cases} \frac { y } { 20 } & y = 1,2 \text { and } 3 \\ \frac { 7 } { 10 } & y = 4 \end{cases}$$ The random variable \(T\) is the value obtained when the number on the face on which it lands is multiplied by 3 . Calculate \(\mathrm { E } ( T )\) and \(\operatorname { Var } ( T )\).
3. Calculate:
   1. \(\mathrm { P } ( X > 1 )\);
   2. \(\mathrm { P } ( X + T \leqslant 9\) and \(X > 1 )\);
   3. \(\mathrm { P } ( X + T \leqslant 9 \mid X > 1 )\).

5

1. Joshua plays a game in which he repeatedly tosses an unbiased coin. His game concludes when he obtains either a head or 5 tails in succession. The random variable \(N\) denotes the number of tosses of his coin required to conclude a game. By completing Table 3 below, calculate \(\mathrm { E } ( N )\).
2. Joshua's sister, Ruth, plays a separate game in which she repeatedly tosses a coin that is biased in such a way that the probability of a head in a single toss of her coin is \(\frac { 1 } { 4 }\). Her game also concludes when she obtains either a head or 5 tails in succession. The random variable \(M\) denotes the number of tosses of her coin required to conclude her game. Complete Table 4 below.
3. 1. Joshua and Ruth play their games simultaneously. Calculate the probability that Joshua and Ruth will conclude their games in an equal number of tosses of their coins.
   2. Joshua and Ruth play their games simultaneously on 3 occasions. Calculate the probability that, on at least 2 of these occasions, their games will be concluded in an equal number of tosses of their coins. Give your answer to three decimal places.
      (4 marks) \begin{table}[h]
      \captionsetup{labelformat=empty} \caption{Table 3}

      \(\boldsymbol { n }\)	1	2	3	4	5
      \(\mathbf { P } ( \boldsymbol { N } = \boldsymbol { n } )\)			\(\frac { 1 } { 8 }\)		\(\frac { 1 } { 16 }\)

      \end{table} \begin{table}[h]
      \captionsetup{labelformat=empty} \caption{Table 4}

      \(\boldsymbol { m }\)	1	2	3	4	5
      \(\mathbf { P } ( \boldsymbol { M } = \boldsymbol { m } )\)	\(\frac { 1 } { 4 }\)	\(\frac { 3 } { 16 }\)

      \end{table}

5 Aiden takes his car to a garage for its MOT test. The probability that his car will need to have \(X\) tyres replaced is shown in the table.

1. Show that the mean of \(X\) is 1.85 and calculate the variance of \(X\).
2. The charge for the MOT test is \(\pounds c\) and the cost of each new tyre is \(\pounds n\). The total amount that Aiden must pay the garage is \(\pounds T\).
   1. Express \(T\) in terms of \(c , n\) and \(X\).
   2. Hence, using your results from part (a), find expressions for \(\mathrm { E } ( T )\) and \(\operatorname { Var } ( T )\).

5 The discrete random variable \(R\) has the following probability distribution.

1. Calculate exact values for \(\mathrm { E } ( R )\) and \(\operatorname { Var } ( R )\).
2. 1. By tabulating the probability distribution for \(X = \frac { 1 } { R ^ { 2 } }\), show that \(\mathrm { E } ( X ) = \frac { 25 } { 64 }\).
   2. Hence find the value of the mean of the area of a rectangle which has sides of length \(\frac { 8 } { R }\) and \(\left( R + \frac { 8 } { R } \right)\).
      (3 marks)

3 Morecrest football team always scores at least one goal but never scores more than four goals in each game. The number of goals, \(R\), scored in each game by the team can be modelled by the following probability distribution.

1. Calculate exact values for the mean and variance of \(R\).
2. Next season the team will play 32 games. They expect to win \(90 \%\) of the games in which they score at least three goals, half of the games in which they score exactly two goals and \(20 \%\) of the games in which they score exactly one goal. Find, for next season:
   1. the number of games in which they expect to score at least three goals;
   2. the number of games that they expect to win.

7

1. The number of text messages, \(N\), sent by Peter each month on his mobile phone never exceeds 40. When \(0 \leqslant N \leqslant 10\), he is charged for 5 messages.
   When \(10 < N \leqslant 20\), he is charged for 15 messages.
   When \(20 < N \leqslant 30\), he is charged for 25 messages.
   When \(30 < N \leqslant 40\), he is charged for 35 messages.
   The number of text messages, \(Y\), that Peter is charged for each month has the following probability distribution:

   \(\boldsymbol { y }\)	5	15	25	35
   \(\mathbf { P } ( \boldsymbol { Y } = \boldsymbol { y } )\)	0.1	0.2	0.3	0.4

   1. Calculate the mean and the standard deviation of \(Y\).
   2. The Goodtime phone company makes a total charge for text messages, \(C\) pence, each month given by $$C = 10 Y + 5$$ Calculate \(\mathrm { E } ( C )\).
2. The number of text messages, \(X\), sent by Joanne each month on her mobile phone is such that $$\mathrm { E } ( X ) = 8.35 \quad \text { and } \quad \mathrm { E } \left( X ^ { 2 } \right) = 75.25$$ The Newtime phone company makes a total charge for text messages, \(T\) pence, each month given by $$T = 0.4 X + 250$$ Calculate \(\operatorname { Var } ( T )\).

4 A discrete random variable \(X\) has the probability distribution $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l } \frac { 3 x } { 40 } & x = 1,2,3,4 \\ \frac { x } { 20 } & x = 5 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Calculate \(\mathrm { E } ( X )\).
2. Show that:
   1. \(\quad \mathrm { E } \left( \frac { 1 } { X } \right) = \frac { 7 } { 20 }\);
      (2 marks)
   2. \(\operatorname { Var } \left( \frac { 1 } { X } \right) = \frac { 7 } { 160 }\).
3. The discrete random variable \(Y\) is such that \(Y = \frac { 40 } { X }\). Calculate:
   1. \(\mathrm { P } ( Y < 20 )\);
   2. \(\mathrm { P } ( X < 4 \mid Y < 20 )\).

4 A house has a total of five bedrooms, at least one of which is always rented.
The probability distribution for \(R\), the number of bedrooms that are rented at any given time, is given by $$\mathrm { P } ( R = r ) = \begin{cases} 0.5 & r = 1 \\ 0.4 ( 0.6 ) ^ { r - 1 } & r = 2,3,4 \\ 0.0296 & r = 5 \end{cases}$$

1. Complete the table below.
2. Find the probability that fewer than 3 bedrooms are not rented at any given time.
3. 1. Find the value of \(\mathrm { E } ( R )\).
   2. Show that \(\mathrm { E } \left( R ^ { 2 } \right) = 4.8784\) and hence find the value of \(\operatorname { Var } ( R )\).
4. Bedrooms are rented on a monthly basis. The monthly income, \(\pounds M\), from renting bedrooms in the house may be modelled by $$M = 1250 R - 282$$ Find the mean and the standard deviation of \(M\).

   \(\boldsymbol { r }\)	1	2	3	4	5
   \(\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )\)	0.5				0.0296

5 In a computer game, players try to collect five treasures. The number of treasures that Isaac collects in one play of the game is represented by the discrete random variable \(X\). The probability distribution of \(X\) is defined by $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l } \frac { 1 } { x + 2 } & x = 1,2,3,4 \\ k & x = 5 \\ 0 & \text { otherwise } \end{array} \right.$$

1. 1. Show that \(k = \frac { 1 } { 20 }\).
   2. Calculate the value of \(\mathrm { E } ( X )\).
   3. Show that \(\operatorname { Var } ( X ) = 1.5275\).
   4. Find the probability that Isaac collects more than 2 treasures.
2. The number of points that Isaac scores for collecting treasures is \(Y\) where $$Y = 100 X - 50$$ Calculate the mean and the standard deviation of \(Y\).

3 A box contains a large number of pea pods. The number of peas in a pod may be modelled by the random variable \(X\). The probability distribution of \(X\) is tabulated below.

1. Two pods are picked randomly from the box. Find the probability that the number of peas in each pod is at most 4.
2. It is given that \(\mathrm { E } ( X ) = 5.1\).
   1. Determine the values of \(a\) and \(b\).
   2. Hence show that \(\operatorname { Var } ( X ) = 1.29\).
   3. Some children play a game with the pods, randomly picking a pod and scoring points depending on the number of peas in the pod. For each pod picked, the number of points scored, \(N\), is found by doubling the number of peas in the pod and then subtracting 5. Find the mean and the standard deviation of \(N\).
      [0pt] [3 marks]

7 Each week, a newsagent stocks 5 copies of the magazine Statistics Weekly. A regular customer always buys one copy. The demand for additional copies may be modelled by a Poisson distribution with mean 2. The number of copies sold in a week, \(X\), has the probability distribution shown in the table, where probabilities are stated correct to three decimal places.

1. Show that, correct to three decimal places, the values of \(a\) and \(b\) are 0.180 and 0.143 respectively.
2. Find the values of \(\mathrm { E } ( X )\) and \(\mathrm { E } \left( X ^ { 2 } \right)\), showing the calculations needed to obtain these values, and hence calculate the standard deviation of \(X\).
3. The newsagent makes a profit of \(\pounds 1\) on each copy of Statistics Weekly that is sold and loses 50 p on each copy that is not sold. Find the mean weekly profit for the newsagent from sales of this magazine.
4. Assuming that the weekly demand remains the same, show that the mean weekly profit from sales of Statistics Weekly will be greater if the newsagent stocks only 4 copies.
   [0pt] [5 marks]
   \includegraphics[max width=\textwidth, alt={}]{6cdf244b-168a-4be5-8ef8-8125daae8608-24_2488_1728_219_141}

7 The random variable \(X\) has a Poisson distribution with parameter \(\lambda\).

1. 1. Prove, from first principles, that \(\mathrm { E } ( X ) = \lambda\).
   2. Hence, given that \(\mathrm { E } ( X ( X - 1 ) ) = \lambda ^ { 2 }\), find, in terms of \(\lambda\), an expression for \(\operatorname { Var } ( X )\).
2. The mode, \(m\), of \(X\) is such that $$\mathrm { P } ( X = m ) \geqslant \mathrm { P } ( X = m - 1 ) \quad \text { and } \quad \mathrm { P } ( X = m ) \geqslant \mathrm { P } ( X = m + 1 )$$
   1. Show that \(\lambda - 1 \leqslant m \leqslant \lambda\).
   2. Given that \(\lambda = 4.9\), determine \(\mathrm { P } ( X = m )\).
3. The random variable \(Y\) has a Poisson distribution with mode \(d\) and standard deviation 15.5. Use a distributional approximation to estimate \(\mathrm { P } ( Y \geqslant d )\).
   \includegraphics[max width=\textwidth, alt={}]{b855b5b3-097e-4894-aaec-d77f515949b0-19_2484_1709_223_153}

7

1. The random variable \(X\) has a Poisson distribution with \(\mathrm { E } ( X ) = \lambda\).
   1. Prove, from first principles, that \(\mathrm { E } ( X ( X - 1 ) ) = \lambda ^ { 2 }\).
   2. Hence deduce that \(\operatorname { Var } ( X ) = \mathrm { E } ( X )\).
2. The random variable \(Y\) has a Poisson distribution with \(\mathrm { E } ( Y ) = 2.5\). Given that \(Z = 4 Y + 30\) :
   1. show that \(\operatorname { Var } ( Z ) = \mathrm { E } ( Z )\);
   2. give a reason why the distribution of \(Z\) is not Poisson.
      \includegraphics[max width=\textwidth, alt={}]{fa3bf9d6-f064-4214-acff-d8b88c33a81e-19_2486_1714_221_153}
      \includegraphics[max width=\textwidth, alt={}]{fa3bf9d6-f064-4214-acff-d8b88c33a81e-20_2486_1714_221_153}

5 The numbers of daily morning operations, \(X\), and daily afternoon operations, \(Y\), in an operating theatre of a small private hospital can be modelled by the following bivariate probability distribution.

1. 1. State why \(\mathrm { E } ( X ) = 4\) and show that \(\operatorname { Var } ( X ) = 0.9\).
   2. Given that $$\mathrm { E } ( Y ) = 3.7 , \operatorname { Var } ( Y ) = 0.61 \text { and } \mathrm { E } ( X Y ) = 14.4$$ calculate values for \(\operatorname { Cov } ( X , Y )\) and \(\rho _ { X Y }\).
2. Calculate values for the mean and the variance of:
   1. \(T = X + Y\);
   2. \(\quad D = X - Y\).
      [0pt] [4 marks]

2 In a quiz, competitors have to match 5 landmarks to the 5 British counties which the landmarks are in. The random variable \(X\) represents the number of correct matches that a competitor gets, assuming that the competitor guesses randomly. The probability distribution of \(X\) is given in the following table.

1. Explain why \(\mathrm { P } ( X = 4 )\) must be 0 .
2. Explain how the value \(\frac { 1 } { 120 }\) for \(\mathrm { P } ( X = 5 )\) is calculated.
3. Draw a graph to illustrate the distribution.
4. Find each of the following.